Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,6,0,0,-7,-6,0,0,-7,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.368464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 185)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.552543\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.180152 q^{2} -1.96755 q^{4} -4.41500 q^{7} -0.714762 q^{8} -4.27171 q^{11} +2.79978 q^{13} -0.795373 q^{14} +3.80632 q^{16} -4.43948 q^{17} +2.43507 q^{19} -0.769559 q^{22} +5.77387 q^{23} +0.504387 q^{26} +8.68672 q^{28} -0.409254 q^{29} +7.79180 q^{31} +2.11524 q^{32} -0.799782 q^{34} +1.00000 q^{37} +0.438683 q^{38} -0.757374 q^{41} -2.19908 q^{43} +8.40479 q^{44} +1.04018 q^{46} -4.26487 q^{47} +12.4922 q^{49} -5.50870 q^{52} -0.137540 q^{53} +3.15568 q^{56} -0.0737281 q^{58} +3.07119 q^{59} -3.02909 q^{61} +1.40371 q^{62} -7.23158 q^{64} +11.4482 q^{67} +8.73487 q^{68} +10.7144 q^{71} -8.20680 q^{73} +0.180152 q^{74} -4.79111 q^{76} +18.8596 q^{77} +7.11193 q^{79} -0.136443 q^{82} -11.3625 q^{83} -0.396170 q^{86} +3.05326 q^{88} +16.2305 q^{89} -12.3610 q^{91} -11.3603 q^{92} -0.768326 q^{94} -18.3399 q^{97} +2.25051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{4} - 7 q^{7} - 6 q^{8} - 7 q^{11} - 2 q^{13} - 4 q^{14} + 8 q^{16} - 8 q^{17} + 14 q^{19} - 2 q^{22} + 2 q^{23} + 20 q^{26} + 14 q^{28} - 2 q^{29} + 8 q^{31} - 22 q^{32} + 12 q^{34} + 5 q^{37}+ \cdots + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.180152 0.127387 0.0636934 − 0.997970i \(-0.479712\pi\)
0.0636934 + 0.997970i \(0.479712\pi\)
\(3\) 0 0
\(4\) −1.96755 −0.983773
\(5\) 0 0
\(6\) 0 0
\(7\) −4.41500 −1.66871 −0.834357 − 0.551224i \(-0.814161\pi\)
−0.834357 + 0.551224i \(0.814161\pi\)
\(8\) −0.714762 −0.252707
\(9\) 0 0
\(10\) 0 0
\(11\) −4.27171 −1.28797 −0.643985 − 0.765038i \(-0.722720\pi\)
−0.643985 + 0.765038i \(0.722720\pi\)
\(12\) 0 0
\(13\) 2.79978 0.776520 0.388260 − 0.921550i \(-0.373076\pi\)
0.388260 + 0.921550i \(0.373076\pi\)
\(14\) −0.795373 −0.212572
\(15\) 0 0
\(16\) 3.80632 0.951581
\(17\) −4.43948 −1.07673 −0.538366 − 0.842711i \(-0.680958\pi\)
−0.538366 + 0.842711i \(0.680958\pi\)
\(18\) 0 0
\(19\) 2.43507 0.558643 0.279321 − 0.960198i \(-0.409890\pi\)
0.279321 + 0.960198i \(0.409890\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.769559 −0.164071
\(23\) 5.77387 1.20393 0.601967 − 0.798521i \(-0.294384\pi\)
0.601967 + 0.798521i \(0.294384\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.504387 0.0989184
\(27\) 0 0
\(28\) 8.68672 1.64164
\(29\) −0.409254 −0.0759967 −0.0379983 − 0.999278i \(-0.512098\pi\)
−0.0379983 + 0.999278i \(0.512098\pi\)
\(30\) 0 0
\(31\) 7.79180 1.39945 0.699724 − 0.714413i \(-0.253306\pi\)
0.699724 + 0.714413i \(0.253306\pi\)
\(32\) 2.11524 0.373926
\(33\) 0 0
\(34\) −0.799782 −0.137161
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0.438683 0.0711638
\(39\) 0 0
\(40\) 0 0
\(41\) −0.757374 −0.118282 −0.0591410 − 0.998250i \(-0.518836\pi\)
−0.0591410 + 0.998250i \(0.518836\pi\)
\(42\) 0 0
\(43\) −2.19908 −0.335357 −0.167679 − 0.985842i \(-0.553627\pi\)
−0.167679 + 0.985842i \(0.553627\pi\)
\(44\) 8.40479 1.26707
\(45\) 0 0
\(46\) 1.04018 0.153366
\(47\) −4.26487 −0.622095 −0.311048 − 0.950394i \(-0.600680\pi\)
−0.311048 + 0.950394i \(0.600680\pi\)
\(48\) 0 0
\(49\) 12.4922 1.78461
\(50\) 0 0
\(51\) 0 0
\(52\) −5.50870 −0.763919
\(53\) −0.137540 −0.0188926 −0.00944630 − 0.999955i \(-0.503007\pi\)
−0.00944630 + 0.999955i \(0.503007\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.15568 0.421695
\(57\) 0 0
\(58\) −0.0737281 −0.00968098
\(59\) 3.07119 0.399835 0.199918 − 0.979813i \(-0.435933\pi\)
0.199918 + 0.979813i \(0.435933\pi\)
\(60\) 0 0
\(61\) −3.02909 −0.387835 −0.193918 − 0.981018i \(-0.562119\pi\)
−0.193918 + 0.981018i \(0.562119\pi\)
\(62\) 1.40371 0.178271
\(63\) 0 0
\(64\) −7.23158 −0.903948
\(65\) 0 0
\(66\) 0 0
\(67\) 11.4482 1.39862 0.699310 − 0.714819i \(-0.253491\pi\)
0.699310 + 0.714819i \(0.253491\pi\)
\(68\) 8.73487 1.05926
\(69\) 0 0
\(70\) 0 0
\(71\) 10.7144 1.27156 0.635781 − 0.771870i \(-0.280678\pi\)
0.635781 + 0.771870i \(0.280678\pi\)
\(72\) 0 0
\(73\) −8.20680 −0.960534 −0.480267 − 0.877122i \(-0.659460\pi\)
−0.480267 + 0.877122i \(0.659460\pi\)
\(74\) 0.180152 0.0209423
\(75\) 0 0
\(76\) −4.79111 −0.549578
\(77\) 18.8596 2.14925
\(78\) 0 0
\(79\) 7.11193 0.800155 0.400077 − 0.916481i \(-0.368983\pi\)
0.400077 + 0.916481i \(0.368983\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.136443 −0.0150676
\(83\) −11.3625 −1.24719 −0.623597 − 0.781746i \(-0.714329\pi\)
−0.623597 + 0.781746i \(0.714329\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.396170 −0.0427201
\(87\) 0 0
\(88\) 3.05326 0.325479
\(89\) 16.2305 1.72043 0.860214 − 0.509933i \(-0.170330\pi\)
0.860214 + 0.509933i \(0.170330\pi\)
\(90\) 0 0
\(91\) −12.3610 −1.29579
\(92\) −11.3603 −1.18440
\(93\) 0 0
\(94\) −0.768326 −0.0792468
\(95\) 0 0
\(96\) 0 0
\(97\) −18.3399 −1.86213 −0.931066 − 0.364850i \(-0.881120\pi\)
−0.931066 + 0.364850i \(0.881120\pi\)
\(98\) 2.25051 0.227336
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8325.2.a.cc.1.3 5
3.2 odd 2 925.2.a.h.1.3 5
5.4 even 2 1665.2.a.q.1.3 5
15.2 even 4 925.2.b.g.149.5 10
15.8 even 4 925.2.b.g.149.6 10
15.14 odd 2 185.2.a.d.1.3 ✓ 5
60.59 even 2 2960.2.a.ba.1.5 5
105.104 even 2 9065.2.a.j.1.3 5
555.554 odd 2 6845.2.a.g.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.d.1.3 ✓ 5 15.14 odd 2
925.2.a.h.1.3 5 3.2 odd 2
925.2.b.g.149.5 10 15.2 even 4
925.2.b.g.149.6 10 15.8 even 4
1665.2.a.q.1.3 5 5.4 even 2
2960.2.a.ba.1.5 5 60.59 even 2
6845.2.a.g.1.3 5 555.554 odd 2
8325.2.a.cc.1.3 5 1.1 even 1 trivial
9065.2.a.j.1.3 5 105.104 even 2